Abstract
In this paper, we obtain some applications of first order differ-
ential subordination and superordination results involving certain linear op-
erator and other linear operators for certain normalized analytic functions.
Some of our results improve and generalize previously known results.
Results & Lemmas (21)
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Lemma 1
Lemma 1 ([16]). Let q(z) be univalent in the unit disk U and θ and ϕ be analytic in a domain D containing q(U) with ϕ(w) ̸= 0 when w ∈q(U).…
Lemma 1 ([16]). Let q(z) be univalent in the unit disk U and θ and ϕ be analytic in a domain D containing q(U) with ϕ(w) ̸= 0 when w ∈q(U). Set (2.1) ψ(z) = zq ′(z)ϕ(q(z)) and h(z) = θ(q(z)) + ψ(z). Suppose that (i) ψ(z) is starlike univalent in U, (ii) ℜ zh ′(z) ψ(z) > 0 for z ∈U. If p(z) is analytic with p(0) = q(0), p(U) ⊂D and
Lemma 2
Lemma 2 ([23]). Let q(z) be univalent in U with q(0) = 1. Let α ∈C, γ ∈C∗, further assume that (2.3) ℜ ( 1 + zq ′′(z) q ′(z) ) > max 0,…
Lemma 2 ([23]). Let q(z) be univalent in U with q(0) = 1. Let α ∈C, γ ∈C∗, further assume that (2.3) ℜ ( 1 + zq ′′(z) q ′(z) ) > max 0, −ℜ α γ
Lemma 3
Lemma 3 ([4]). Let q(z) be convex univalent in U and ϑ and φ be analytic in a domain D containing q(U). Suppose that (i) ℜ ϑ ′(q(z))…
Lemma 3 ([4]). Let q(z) be convex univalent in U and ϑ and φ be analytic in a domain D containing q(U). Suppose that (i) ℜ ϑ ′(q(z)) φ(q(z)) > 0 for z ∈U, (ii) Ψ(z) = zq ′(z)φ(q(z)) is starlike univalent in U. If p(z) ∈H[q(0), 1] ∩Q, with p(U) ⊆D, and ϑ(p(z)) + zp ′(z)φ(p(z)) is univalent in U and (2.4)
Lemma 4
Lemma 4 ([23]). Let q(z) be convex univalent in U, q(0) = 1. Let α ∈C, γ ∈C∗and ℜ α γ > 0. If p(z) ∈H[q(0), 1]∩Q, αp(z)+γzp ′(z) is…
Lemma 4 ([23]). Let q(z) be convex univalent in U, q(0) = 1. Let α ∈C, γ ∈C∗and ℜ α γ > 0. If p(z) ∈H[q(0), 1]∩Q, αp(z)+γzp ′(z) is univalent in U and αq(z) + γzq ′(z) ≺αp(z) + γzp ′(z), then q(z) ≺p(z) and q(z) is the best subordinant. 3. Sandwich results. Unless otherwise mentioned, we assume throughout this paper that λ > 0 and n ∈N0.
Theorem 1.
Theorem 1. Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Further, assume that (3.1) ℜ ( 1 + zq ′′(z) q ′(z) ) > max 0, −ℜ 1 γ
Theorem 1. Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Further, assume that (3.1) ℜ ( 1 + zq ′′(z) q ′(z) ) > max 0, −ℜ 1 γ
Corollary 1.
Corollary 1. Let γ ∈C∗and ℜ 1 −Bz 1 + Bz > max 0, −ℜ 1 γ . If f, g ∈A satisfy the following subordination condition: Dn λ(f ∗g)(z)
Corollary 1. Let γ ∈C∗and ℜ 1 −Bz 1 + Bz > max 0, −ℜ 1 γ . If f, g ∈A satisfy the following subordination condition: Dn λ(f ∗g)(z)
Corollary 2
Corollary 2 ([17, Corollary 7]). Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Further assume that (3.1) holds. If f ∈A satisfies the…
Corollary 2 ([17, Corollary 7]). Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Further assume that (3.1) holds. If f ∈A satisfies the following subordination condition: Dnf(z) Dn+1f(z) + γ 1 −Dnf(z)Dn+2f(z) [Dn+1f(z)]2 ≺q(z) + γzq ′(z), then Dnf(z) Dn+1f(z) ≺q(z) and q(z) is the best dominant.
Corollary 3.
Corollary 3. Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Fur- ther assume that (3.1) holds. If f ∈A satisfies the following…
Corollary 3. Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Fur- ther assume that (3.1) holds. If f ∈A satisfies the following subordination condition: (1−γ) Hl,m (a1; b1) f(z) z (Hl,m (a1; b1) f(z)) ′ + γ ( 1−Hl,m (a1; b1) f(z) (Hl,m (a1; b1) f(z)) ′′ (Hl,m (a1; b1) f(z)) ′2 ) ≺q(z) + γzq
Corollary 4.
Corollary 4. Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Fur- ther assume that (3.1) holds. If f ∈A satisfies the following…
Corollary 4. Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Fur- ther assume that (3.1) holds. If f ∈A satisfies the following subordination condition: Dn λ(a1; b1)f(z) Dn+1 λ (a1; b1)f(z) + γ λ ( 1 −Dn λ(a1; b1)f(z)Dn+2 λ (a1; b1)f(z)
Corollary 5.
Corollary 5. Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Fur- ther assume that (3.1) holds. If f ∈A satisfies the following…
Corollary 5. Let q(z) be univalent in U with q(0) = 1, and γ ∈C∗. Fur- ther assume that (3.1) holds. If f ∈A satisfies the following subordination condition: (1 −γ) I (s, l) f(z) z (I (s, l) f(z)) ′ + γ ( 1 −I (s, l) f(z) (I (s, l) f(z)) ′′ (I (s, l) f(z)) ′2 ) ≺q(z) + γzq
Theorem 2.
Theorem 2. Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f, g ∈A such that Dn λ(f∗g)(z) Dn+1 λ (f∗g)(z) ∈H…
Theorem 2. Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f, g ∈A such that Dn λ(f∗g)(z) Dn+1 λ (f∗g)(z) ∈H [1, 1] ∩Q, Dn λ(f ∗g)(z) Dn+1 λ (f ∗g)(z) + γ λ ( 1 −Dn
Corollary 6.
Corollary 6. Let γ ∈C with ℜ(¯γ) > 0. If f, g ∈A such that Dn λ(f∗g)(z) Dn+1 λ (f∗g)(z) ∈ H [1, 1] ∩Q, Dn λ(f ∗g)(z) Dn+1 λ (f ∗g)(z) + γ λ…
Corollary 6. Let γ ∈C with ℜ(¯γ) > 0. If f, g ∈A such that Dn λ(f∗g)(z) Dn+1 λ (f∗g)(z) ∈ H [1, 1] ∩Q, Dn λ(f ∗g)(z) Dn+1 λ (f ∗g)(z) + γ λ ( 1 −Dn
Corollary 7
Corollary 7 ([17, Corollary 12]). Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f ∈A such that Dnf(z)…
Corollary 7 ([17, Corollary 12]). Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f ∈A such that Dnf(z) Dn+1f(z) ∈ H [1, 1] ∩Q, Dnf(z) Dn+1f(z) + γ 1 −Dnf(z)Dn+2f(z) [Dn+1f(z)]2 is univalent in U, and the following superordination condition q(z) + γzq
Corollary 8.
Corollary 8. Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f ∈A such that Hl,m(a1;b1)f(z) z(Hl,m(a1;b1)f(z))…
Corollary 8. Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f ∈A such that Hl,m(a1;b1)f(z) z(Hl,m(a1;b1)f(z)) ′ ∈H [1, 1] ∩Q, (1 −γ) Hl,m (a1; b1) f(z) z (Hl,m (a1; b1) f(z)) ′ + γ ( 1 −Hl,m (a1; b1) f(z) (Hl,m (a1; b1) f(z)) ′′ (Hl,m (a1; b1) f(z))
Corollary 9.
Corollary 9. Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f, g ∈A such that Dn λ(a1;b1)f(z) Dn+1 λ…
Corollary 9. Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f, g ∈A such that Dn λ(a1;b1)f(z) Dn+1 λ (a1;b1)f(z) ∈H [1, 1] ∩Q, Dn λ(a1; b1)f(z) Dn+1 λ (a1; b1)f(z) + γ λ ( 1 −Dn
Corollary 10.
Corollary 10. Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f ∈A such that I(s,l) z(I(s,l)f(z))′ ∈H [1, 1]…
Corollary 10. Let q(z) be convex univalent in U with q(0) = 1. Let γ ∈C with ℜ(¯γ) > 0. If f ∈A such that I(s,l) z(I(s,l)f(z))′ ∈H [1, 1] ∩Q, (1 −γ) I(s, l)f(z) z (I(s, l)f(z)) ′ + γ ( 1 −I(s, l)f(z) (I(s, l)f(z)) ′′ (I(s, l)f(z)) ′2 )
Theorem 3.
Theorem 3. Let q1(z) be convex univalent in U with q1(0) = 1, γ ∈C with ℜ(¯γ) > 0, q2(z) be univalent in U with q2(0) = 1, and satisfy…
Theorem 3. Let q1(z) be convex univalent in U with q1(0) = 1, γ ∈C with ℜ(¯γ) > 0, q2(z) be univalent in U with q2(0) = 1, and satisfy (3.1). If f, g ∈A such that Dn λ(f∗g)(z) Dn+1 λ (f∗g)(z) ∈H [1, 1] ∩Q, Dn λ(f ∗g)(z) Dn+1 λ (f ∗g)(z) + γ λ (
Theorem 3
Theorem 3, we obtain the following corollary.
Theorem 3, we obtain the following corollary.
Corollary 11.
Corollary 11. Let γ ∈C with ℜ(¯γ) > 0. If f, g ∈A such that Dn λ(f∗g)(z) Dn+1 λ (f∗g)(z) ∈ H [1, 1] ∩Q, Dn λ(f ∗g)(z) Dn+1 λ (f ∗g)(z) + γ…
Corollary 11. Let γ ∈C with ℜ(¯γ) > 0. If f, g ∈A such that Dn λ(f∗g)(z) Dn+1 λ (f∗g)(z) ∈ H [1, 1] ∩Q, Dn λ(f ∗g)(z) Dn+1 λ (f ∗g)(z) + γ λ ( 1 −Dn
Corollary 12.
Corollary 12. Let q1(z) be convex univalent in U with q1(0) = 1, γ ∈C with ℜ(¯γ) > 0, q2(z) be univalent in U with q2(0) = 1, and satisfy…
Corollary 12. Let q1(z) be convex univalent in U with q1(0) = 1, γ ∈C with ℜ(¯γ) > 0, q2(z) be univalent in U with q2(0) = 1, and satisfy (3.1). If f ∈A such that Dnf(z) Dn+1f(z) ∈H [1, 1] ∩Q, Dnf(z) Dn+1f(z) + γ 1 −Dnf(z)Dn+2f(z) [Dn+1f(z)]2 is univalent in U, and q1(z) + γzq ′ 1(z) ≺
Corollary 10
Corollary 10, we obtain similar sandwich theorems for the corresponding linear operators.
Corollary 10, we obtain similar sandwich theorems for the corresponding linear operators.
Function classes studied:
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